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On stochastic porous-medium equations with critical-growth conservative multiplicative noise
Best approximation of orbits in iterated function systems
1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
2. | School of Science, Wuhan University of Technology, Wuhan 430074, China |
$ \Phi = \{\phi_{i}\colon i\in\Lambda\} $ |
$ (X,d) $ |
$ \Lambda = \{1, 2, \ldots,l\} $ |
$ l \ge2 $ |
$ \Lambda = \{1,2,\ldots\} $ |
$ J $ |
$ \Phi $ |
$ D $ |
$ x\in J\backslash D, $ |
$ \{\omega_{n}(x)\}_{n\geq 1}\subset \Lambda^{\mathbb{N}} $ |
$ x, $ |
$ \{x\} = \bigcap\limits_n\phi_{\omega_{1}(x)}\circ\cdots\circ\phi_{\omega_{n}(x)}(X). $ |
$ x = [\omega_{1}(x),\omega_{2}(x),\ldots]. $ |
$ x, y\in J\backslash D, $ |
$ M_{n}(x,y) $ |
$ \begin{align*} M_{n}(x,y) = \max\big\{k\in \mathbb{N}\colon \omega_{i+1}(x) = \omega_{i+1}(y),\ldots, &\omega_{i+k}(x) = \omega_{i+k}(y) \; \\&\text{for some}\; 0\leq i \leq n-k\big\}, \end{align*} $ |
$ n $ |
$ (x,y). $ |
$ M_{n}(x,y) $ |
$ n $ |
$ \infty. $ |
$ N $ |
References:
[1] |
M. D. Boshernitzan,
Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.
doi: 10.1007/bf01244320. |
[2] |
D. Bessis, G. Paladin, G. Turchetti and S. Vaienti,
Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, J. Statist. Phys., 51 (1988), 109-134.
doi: 10.1007/bf01015323. |
[3] |
L. Barreira and B. Saussol,
Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463.
doi: 10.1007/s002200100427. |
[4] |
Y. Bugeaud and B.-W. Wang,
Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions, J. Fractal Geom., 1 (2014), 221-241.
doi: 10.4171/jfg/6. |
[5] |
I. P. Coornfeld, S. V. Fomin and Ya. G. Sinaǐ, Ergodic Theory, Springer-Verlag, New York, 1982. |
[6] |
N. Chernov and D. Kleinbock,
Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.
doi: 10.1007/bf02809888. |
[7] |
K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^nd$ edition, John Wiley & Sons, Ltd., Chichester, 2014. |
[8] |
J. L. Fernández, M. V. Melián and D. Pestana, Quantitative recurrence properties of expanding maps, preprint, arXiv: math/0703222. Google Scholar |
[9] |
S. Galatolo,
Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.
doi: 10.4310/mrl.2007.v14.n5.a8. |
[10] |
G. H. Hardy and E. M. Wright, An Introduction to The Theory of Numbers, 5$^nd$ edition, The Clarendon Press, Oxford University Press, New York, 1979.
![]() |
[11] |
R. Hill and S. L. Velani,
The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.
doi: 10.1007/bf01245179. |
[12] |
R. Hill and S. L. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., (1997), 193–216.
doi: 10.1007/bf02699537. |
[13] |
N. Haydn and S. L. Vaienti,
The Rényi entropy function and the large deviation of short return times, Ergodic Theory Dynam. Systems., 30 (2010), 159-179.
doi: 10.1017/s0143385709000030. |
[14] |
H. Jager and C. de Vroedt,
Lüroth series and their ergodic properties, Nederl. Akad. Wetensch. Proc. Ser., 31 (1969), 31-42.
doi: 10.1016/1385-7258(69)90023-7. |
[15] |
A. Ya. Khintchine, Continued Fractions, Translated by Peter WynnP. Noordhoff, Ltd., Groningen 1963.
doi: 10.1017/s0008439500032033. |
[16] |
B. Li, B. W. Wang, J. Wu and J. Xu,
The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc., 108 (2014), 159-186.
doi: 10.1112/plms/pdt017. |
[17] |
J. Li and X. Yang,
On longest matching consecutive subsequence, Int. J. Number Theory., 15 (2019), 1745-1758.
doi: 10.1142/s1793042119500970. |
[18] |
R. D. Mauldin and M. Urbański,
Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.
doi: 10.1112/plms/s3-73.1.105. |
[19] |
R. D. Mauldin and M. Urbański,
Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.
doi: 10.1090/s0002-9947-99-02268-0. |
[20] |
D. S. Ornstein and B. Weiss,
Entropy and data compression schemes, IEEE Trans. Inform. Theory., 39 (1993), 78-83.
doi: 10.1109/18.179344. |
[21] |
L. Peng,
On the hitting depth in the dynamical system of continued fractions, Chaos. Solitons. Fractals., 69 (2014), 22-30.
doi: 10.1016/j.chaos.2014.09.003. |
[22] |
L. Peng, B. Tan and B. W. Wang,
Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.
doi: 10.1007/s11425-011-4303-9. |
[23] |
B. Saussol,
Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267.
doi: 10.3934/dcds.2006.15.259. |
[24] |
B. O. Stratmann and M. Urbánski,
Jarník, Julia$\colon$a Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, Math. Scand., 91 (2002), 27-54.
doi: 10.7146/math.scand.a-14377. |
[25] |
S. Seuret and B.-W. Wang,
Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280 (2015), 472-505.
doi: 10.1016/j.aim.2015.02.019. |
[26] |
B. Tan and B.-W. Wang,
Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.
doi: 10.1016/j.aim.2011.06.034. |
[27] |
F. Takens and E. Verbitski,
Generalized entropies$\colon$Rényi and correlation integral approach, Nonlinearity., 11 (1998), 771-782.
doi: 10.1088/0951-7715/11/4/001. |
[28] |
B. Tan and Q. L. Zhou,
The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.
doi: 10.1016/j.jmaa.2019.05.029. |
[29] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[30] |
Q.-L. Zhou,
Dimensions of recurrent sets in $\beta$-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762-1776.
doi: 10.1016/j.jmaa.2018.12.022. |
show all references
References:
[1] |
M. D. Boshernitzan,
Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.
doi: 10.1007/bf01244320. |
[2] |
D. Bessis, G. Paladin, G. Turchetti and S. Vaienti,
Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets, J. Statist. Phys., 51 (1988), 109-134.
doi: 10.1007/bf01015323. |
[3] |
L. Barreira and B. Saussol,
Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463.
doi: 10.1007/s002200100427. |
[4] |
Y. Bugeaud and B.-W. Wang,
Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$-expansions, J. Fractal Geom., 1 (2014), 221-241.
doi: 10.4171/jfg/6. |
[5] |
I. P. Coornfeld, S. V. Fomin and Ya. G. Sinaǐ, Ergodic Theory, Springer-Verlag, New York, 1982. |
[6] |
N. Chernov and D. Kleinbock,
Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math., 122 (2001), 1-27.
doi: 10.1007/bf02809888. |
[7] |
K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3$^nd$ edition, John Wiley & Sons, Ltd., Chichester, 2014. |
[8] |
J. L. Fernández, M. V. Melián and D. Pestana, Quantitative recurrence properties of expanding maps, preprint, arXiv: math/0703222. Google Scholar |
[9] |
S. Galatolo,
Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805.
doi: 10.4310/mrl.2007.v14.n5.a8. |
[10] |
G. H. Hardy and E. M. Wright, An Introduction to The Theory of Numbers, 5$^nd$ edition, The Clarendon Press, Oxford University Press, New York, 1979.
![]() |
[11] |
R. Hill and S. L. Velani,
The ergodic theory of shrinking targets, Invent. Math., 119 (1995), 175-198.
doi: 10.1007/bf01245179. |
[12] |
R. Hill and S. L. Velani, Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., (1997), 193–216.
doi: 10.1007/bf02699537. |
[13] |
N. Haydn and S. L. Vaienti,
The Rényi entropy function and the large deviation of short return times, Ergodic Theory Dynam. Systems., 30 (2010), 159-179.
doi: 10.1017/s0143385709000030. |
[14] |
H. Jager and C. de Vroedt,
Lüroth series and their ergodic properties, Nederl. Akad. Wetensch. Proc. Ser., 31 (1969), 31-42.
doi: 10.1016/1385-7258(69)90023-7. |
[15] |
A. Ya. Khintchine, Continued Fractions, Translated by Peter WynnP. Noordhoff, Ltd., Groningen 1963.
doi: 10.1017/s0008439500032033. |
[16] |
B. Li, B. W. Wang, J. Wu and J. Xu,
The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc., 108 (2014), 159-186.
doi: 10.1112/plms/pdt017. |
[17] |
J. Li and X. Yang,
On longest matching consecutive subsequence, Int. J. Number Theory., 15 (2019), 1745-1758.
doi: 10.1142/s1793042119500970. |
[18] |
R. D. Mauldin and M. Urbański,
Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc., 73 (1996), 105-154.
doi: 10.1112/plms/s3-73.1.105. |
[19] |
R. D. Mauldin and M. Urbański,
Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025.
doi: 10.1090/s0002-9947-99-02268-0. |
[20] |
D. S. Ornstein and B. Weiss,
Entropy and data compression schemes, IEEE Trans. Inform. Theory., 39 (1993), 78-83.
doi: 10.1109/18.179344. |
[21] |
L. Peng,
On the hitting depth in the dynamical system of continued fractions, Chaos. Solitons. Fractals., 69 (2014), 22-30.
doi: 10.1016/j.chaos.2014.09.003. |
[22] |
L. Peng, B. Tan and B. W. Wang,
Quantitative Poincaré recurrence in continued fraction dynamical system, Sci. China Math., 55 (2012), 131-140.
doi: 10.1007/s11425-011-4303-9. |
[23] |
B. Saussol,
Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267.
doi: 10.3934/dcds.2006.15.259. |
[24] |
B. O. Stratmann and M. Urbánski,
Jarník, Julia$\colon$a Diophantine analysis for geometrically finite Kleinian groups with parabolic elements, Math. Scand., 91 (2002), 27-54.
doi: 10.7146/math.scand.a-14377. |
[25] |
S. Seuret and B.-W. Wang,
Quantitative recurrence properties in conformal iterated function systems, Adv. Math., 280 (2015), 472-505.
doi: 10.1016/j.aim.2015.02.019. |
[26] |
B. Tan and B.-W. Wang,
Quantitative recurrence properties for beta-dynamical system, Adv. Math., 228 (2011), 2071-2097.
doi: 10.1016/j.aim.2011.06.034. |
[27] |
F. Takens and E. Verbitski,
Generalized entropies$\colon$Rényi and correlation integral approach, Nonlinearity., 11 (1998), 771-782.
doi: 10.1088/0951-7715/11/4/001. |
[28] |
B. Tan and Q. L. Zhou,
The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., 478 (2019), 229-235.
doi: 10.1016/j.jmaa.2019.05.029. |
[29] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[30] |
Q.-L. Zhou,
Dimensions of recurrent sets in $\beta$-symbolic dynamics, J. Math. Anal. Appl., 472 (2019), 1762-1776.
doi: 10.1016/j.jmaa.2018.12.022. |
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